Carbon dioxide (CO2) storage in deep geological formations can lead to significant reductions in anthropogenic CO2 emissions if large amounts of CO2 can be stored. Estimates of the storage capacity are therefore essential to the evaluation of individual storage sites as well as the feasibility of the technology. One important limitation on the storage capacity is the radius of review, the lateral extent of the pressure perturbation, of the storage project. We show that pressure dissipation into ambient mudrocks retards lateral pressure propagation significantly and therefore increases the storage capacity. For a three-layer model of a reservoir surrounded by thick mudrocks, the far-field pressure is approximated well by a single-phase model. Through dimensional analysis and numerical simulations, we show that the lateral extent of the pressure front follows a power law that depends on a single dissipation parameter , where and are the ratios of mudrock to reservoir permeability and specific storage, and is the aspect ratio of the confined pressure plume. Both the coefficient and the exponent of the power law are sigmoid decreasing functions of . The values of typical storage sites are in the region where the power-law changes rapidly. The combination of large uncertainty in mudrock properties and the sigmoid shape leads to wide and strongly skewed probability distributions for the predicted radius of review and storage capacity. Therefore, if the lateral extent of the pressure front limits the storage capacity, the determination of the mudrock properties is an important component of site characterization.
 Geological storage of has been proposed in saline aquifers, deep sea sediments, depleted oil and gas reservoirs, and in combination with enhanced oil recovery or enhanced coalbed methane recovery [Intergovernmental Panel on Climate Change (IPCC), 2005]. To make a significant contribution to the mitigation of climate change, gigatonnes of must be stored in the subsurface every year, and therefore, large regional saline aquifers are the primary target for geological storage [Metz et al., 2006].
 To decrease the storage volume, is injected at depths greater than 800 m, where is supercritical for most geotherms [Holloway and Savage, 1993]. Supercritical is less dense than the brine under most continental and shallow marine storage conditions [Bachu, 2008], and therefore, a low-porosity and low-permeability seal that provides a capillary entry barrier for the is necessary to prevent direct upward migration of buoyant supercritical . Evidence that a typical seal has blocked upward migration of the injected out of the storage formation is provided by repeated seismic surveys at the Sleipner -injection site [Chadwick et al., 2005].
 The viability of a storage project hinges on its storage capacity, the amount of that can be injected into the storage formation. Many studies have been carried out to assess the capacity of sedimentary basins to store [Myer et al., 2005; Bradshaw et al., 2007; Kopp et al., 2009]. The estimate of a basin-scale storage capacity is based on the effective pore volume of the target formation [Bachu and Adams, 2003], but the storage capacity is also affected by heterogeneity, residual saturation of each phase, pressure buildup, migration, and the injection scenario [Bachu, 2008; Hesse et al., 2008]. This study is focused on the limitation due to pressure buildup in the reservoir. For the purpose of this study, the pressure buildup, or the overpressure, is defined as the increase of the reservoir pressure above the preinjection pressure. Thibeau and Mucha  argue that either local or global pressure buildup is the primary control on the storage capacity and that a simple volumetric capacity assessment overestimates the storage capacity. Szulczewski et al.  argue that pressure buildup is the limiting factor for short injection times while migration would be limiting for longer injection times.
 Hydraulically closed formations are not suitable for large injection volumes [Ehlig-Economides and Economides, 2010], which require open, permeable, thick, and laterally continuous regional aquifers to allow the displacement of in situ brine by the injected [Van der Meer, 1992; Holloway et al., 1996; Doughty and Pruess, 2004]. Even in open formations, the storage capacity may be limited by local pressure buildup near wells or by basin-scale (10–100 km) pressure buildup [Thibeau and Mucha, 2011]. Geomechanical studies show that low injectivity leads to local pressure buildup near the injector that can cause the geomechanical failure of the surrounding seals [Rutqvist and Tsang, 2002; Rutqvist et al., 2007; Birkholzer et al., 2009; Zhou et al., 2009; Rutqvist et al., 2010; Morris et al., 2011; Vilarrasa et al., 2011]. The limit placed on the injection operation due to local pressure buildup could be mitigated by the addition of injectors that reduce the maximum pressure by distributing the injected over a larger area. In contrast, Thibeau and Mucha  point out that regional scale pressure buildup resulting from the displacement of formation fluids cannot be mitigated by the addition of injectors within the same field. Figure 1 shows the comparison of the pressure perturbation induced by either a single or multiple well(s), which inject the same total amount of fluid, into an identical reservoir with properties given in Table 1. In the scenario with multiple injectors, the maximum of pressure buildup at the center is greatly reduced relative to the single injector scenario, but the radial extent of the pressure buildup is nearly identical. This illustrates that it is easier to manage the local pressure buildup with optimal design of the injection operation than to control its regional extent, which mainly reflects the total fluid mass injected (similar ). Recently, it has been recognized that the radial extent of the pressure perturbation may therefore be an important constraint on the overall storage capacity [Birkholzer and Zhou, 2009; Birkholzer et al., 2011a] and requires the definition of a radius of review for a geological storage site. To establish a radius of review , a pressure cutoff has to be defined. The area inside the radius of review, where the pressure increases above this cutoff, is considered to be affected by the injection operation. Physical processes that have been considered in the definition of the pressure cutoff are brine displacement, capillary entry pressure, and geomechanical failure.
If a parameter was varied systemically, it is indicated by “Var.”, and given in the legend or axes of the respective figure.
10, 25, 50, 75, 100
Nicot et al.  argue that the pressure cutoff for the radius (area) of review of geological storage should be determined based on the possibility of contaminating nearby potable aquifers. This is similar to an earlier definition developed by Thornhill et al. , which defines the radius of review as the area in which injection-induced overpressure may cause migration of the injected or preexisting formation fluids into potable groundwater resources. Nicot et al.  and Birkholzer et al. [2011b] argue that such leakage is most likely through a permeable conduit, i.e., a borehole or a preexisting fracture. They define the pressure cutoff as the minimum value above which sustained migration of formation fluids into the potable aquifer is induced. Based on a static mass balance, they determine a pressure cutoff in the range from 0.1 to 0.6 MPa for a typical reservoir at a depth of 1.5 km. Their approach assumes that the sealing unit contains a preexisting permeable conduit, which lowers the pressure cutoff compared to an intact seal. The other constraint can be the capillary entry pressure into the sealing unit, which is a gradient of 1.9 MPa/km [Springer and Lindgren, 2006]. One geomechanical aspect is that the hydraulic fracturing of the sealing unit can limit the maximum injection-induced overpressure ranging between 3.5 and 8 MPa/km [Thibeau and Mucha, 2011]. The geomechanical and capillary entry pressure constraints can be addressed by a suitable design of the injection scenario, and here we focus on the pressure cutoff imposed by potential brine displacement. Therefore, the lower pressure cutoffs for brine displacement are relevant for the definition of the radius of review. Here we choose a pressure cutoff of 1 MPa, somewhat larger than the values cited above, which consider the extreme case of an open leakage path. Section 2.4 shows that the definition of the radius of review is not sensitive to the absolute value of the pressure cutoff, as long as it is a small fraction of the characteristic injection pressure defined in section 2.3. The absolute value chosen for the pressure cutoff therefore does not affect the results of this study.
 It is common for storage studies to focus only on the target formation and to exclude ambient mudrocks, defined as a fine- to very fine-grained siliciclastic sediments or sedimentary rocks [Grainger, 1984] bounding the reservoir because low-permeability mudrocks prevent vertical migration due to high capillary entry pressure. However, most regional aquifers are not closed and also the overlying and underlying mudrocks are not perfectly impermeable [Neuzil, 1994; Dewhurst et al., 1999]. Pressure buildup caused by injection may partially dissipate into and through these units even if the capillary entry pressure does not permit flow across the boundary as shown in Figure 2. The vertical pressure communication between layers mostly depends on the vertical permeability and specific storage of the seals [Domenico and Schwartz, 1998; Hovorka et al., 2001; Hart et al., 2006; Zhou et al., 2008; Birkholzer et al., 2009; Chadwick et al., 2009; Szulczewski et al., 2012].
Neuzil  shows that both laboratory and regional studies give the effective permeability of the sealing unit between and , and small-scale fractures may increase the permeability. However, in many realistic field sites, the seal will not be a uniform mudrock. This study was partly motivated by the Cranfield site, where the sealing unit is sedimentologically heterogeneous with complex and fine interlayering of laterally discontinuous mudrocks and coarser sediments [Lu et al., 2011]. Because spatial distribution of these fine details are neither known nor is it possible to resolve them numerically, they have to be represented by increased effective mudrock permeability.
 The second important physical property affecting the pore fluid pressure is the specific storage of the system due to the rock and fluid compressibilities. The relatively few data for the specific storage at greater depth are reviewed below. The hysteretic specific storage has to be kept in mind when the unloading during geological storage is compared with the loading of the aquifer during water withdrawal. The impact of hysteresis on the specific storage decreases with increasing depth of burial and we are not aware of data that constrain it at depth. Therefore, the hysteretic effect is neglected in this analysis, but the likely effect of hysteresis on the results is discussed in section 5.2.
Rieke and Chilingarian  obtain compressibility equations as a function of pressure that shows that clay-rich mudrocks have compressibilities up to 2 orders of magnitude higher than fluids at the depth of up to 3 km for storage as well as 1–3 orders of magnitude higher than the reservoir rock itself, which ranges from to at depths greater than 1 km [Ge and Garven, 1992]. This suggests that the ambient permeable and compressible mudrock may contribute significantly to the dissipation of the injection pressure as shown in Figure 2.
 Simulations of geological storage that include the ambient mudrocks often assume they have small compressibilities and negligible permeabilities to avoid a significant impact on the pressure evolution in the reservoir [Rutqvist and Tsang, 2002; Audigane et al., 2007; Van der Meer and Van Wees, 2006; Rutqvist et al., 2007; Vilarrasa et al., 2011]. In contrast, many hydrological studies consider the pressure response in layers adjacent to the pumped aquifer, and they show that groundwater withdrawal from a confined aquifer induces a significant pressure perturbation in the pumped aquifer as well as adjacent mudrocks [Wolff, 1970; Hsieh, 1996; Leake and Hsieh, 1997; Burbey, 2001; Muggeridge et al., 2005]. Table 2 summarizes the petrophysical properties of ambient mudrocks used in previous modeling studies of geological storage and some hydrological studies for comparison. Despite the differences in aquifer depth and loading conditions, the range of mudrock specific storage used in the literature of geological storage is broadly comparable with the hydrological studies, but the range of mudrock permeabilities assumed in studies of geological storage is significantly lower.
Table 2. Physical Properties of the Sandstone Reservoir and Ambient Mudrock for the Previous Studies
Properties in the first five literature are used in the hydrological models, and those in bottom 14 literature are used in the storage models.
Drained vertical bulk modulus defined as .
Mudrock compressibility of a Frio-type reservoir is from Riney et al. .
Mudrock compressibility of a Cranfield-type reservoir is from Chierici et al. .
 In section 2, we present a dimensional analysis that shows that the ratios of permeabilities and specific storage of the sandstone reservoir and the ambient mudrocks ( and ) are the governing parameters determining the pressure evolution in a layered formation. Figure 3 shows the values of and used in the previous studies of hydraulic pumping and storage. We can see that the ratios and for storage sites vary by 6 and 3 orders of magnitude, respectively. This suggests that current injection sites shown in Figure 3 span the range of almost no dissipation of overpressure to a considerable dissipation into the mudrocks and may undergo a large range of possible reservoir responses to injection-induced overpressure. In this manuscript, we present a comprehensive analysis of the effect of pressure dissipation into ambient mudrocks on the lateral pressure propagation, the radius of review, and the storage capacity. In particular, we highlight the role of the mudrock-specific storage, which can be significantly different from that of the reservoir ( ).
 For the purpose of this study, we model the rock volume in which injection is taking place as an infinite horizontal sandstone layer overlain by infinitely thick mudrocks that exhibit various combinations of permeability ( to ) and specific storage ( to ) given in Table 1. We perform a scaling analysis and a simulation study of this simplified layered model to provide a pressure history throughout the storage formation as well as the surrounding mudrocks. Hence, we calculate the radius of review, the storage capacity, and the volume fraction of the brine displaced into the ambient mudrock as a function of the petrophysical parameters of both layers, the thickness of the reservoir, and the injection rate and duration.
2. Model Problem
 The injection of into a brine-filled reservoir leads to a two-phase flow near the injector. Nicot et al.  argue that at late times and large distances the pressure disturbance created by the single- and two-phase injection will be similar. Figure 4 shows that the pressure profiles from single- and two-phase flow in radial geometry match well outside the region directly invaded by the . This confirms that single-phase pressure is a good approximation for the pressure evolution beyond the region under the effect of two-phase flow. However, the pressure at the well is significantly lower due to the increase in total mobility with increasing saturation. Here we focus on regional-scale pressure necessary to define the radius of review and therefore neglect two-phase flow near the injection well.
2.1. Governing Equations
 The equation for pore pressure dissipation in single-phase flow in a porous medium is obtained from the equation of mass conservation and Darcy's law. Assuming constant external stresses and uniaxial deformation in the vertical direction, we study the pressure evolution using a single-phase pressure diffusion model. In cylindrical coordinates, , centered on the injection well, the flow of a slightly compressible fluid in a heterogeneous and isotropic compressible porous medium is described by the diffusion equation
and the initial condition
where is the overpressure defined as the amount of pore pressure exceeding the hydrostatic pressure is the total compressibility, is the permeability, is the porosity, and (Pa s) is the fluid viscosity. Under the assumptions of incompressible solid grains and pores as well as uniaxial strain, the total compressibility is defined as
where (Pa−1) and (Pa−1) are the fluid and rock bulk compressibilities, respectively [Van der Kamp and Gale, 1983; Green and Wang, 1990]. For the range of data given in Table 2, the rock compressibilities for both sandstone and mudrock are several orders of magnitude larger than the fluid compressibility, which we assume to be constant and equal to Pa−1 [Freeze and Cherry, 1979]. The left side of equation (1) suggests the introduction of the uniaxial specific storage S (Pa ) given by Wang .
 An interesting result of this approximation of the total compressibility and the small fluid compressibility is that the specific storage S is essentially independent of porosity. The model properties are summarized in Table 1, and the reference cases for the application of the results are defined using physical properties from the injection sites: In Salah (Krechba, Algeria), Frio (TX, USA), and Cranfield (MS, USA). To facilitate comparison, we are considering a generic injection scenario that does not reproduce the particular conditions at either one of these sites; therefore, we refer to them below as In Salah-type, Frio-type, and Cranfield-type sites or reservoirs.
2.2. Layered Mudrock-Sandstone Geometry
 To study the effect of pressure dissipation into ambient formations from a single well, we consider a three-layer geometry comprising a laterally extensive horizontal sandstone reservoir overlain and underlain by thick mudrocks, assumed to be thick enough to contain the entire vertical extent of the pressure plume, and the domain is therefore radially and vertically infinite. We use a cylindrical coordinate system with the origin at the location of the injector and centered on the sandstone reservoir. In this geometry, overpressure is symmetric across a horizontal plane at the center of the reservoir so that only the positive z axis has to be considered (Figure 5). Assuming fluid injection at with constant rate Q (m /s), the boundary conditions are given by
where (m) is the half thickness of the sandstone reservoir in z direction and (yrs) is the injection period. The spatial variation of the physical properties in this layered geometry is given by
where the subscripts s and m denote sandstone and mudrock, respectively.
2.3. Scaling Analysis
 To develop a dimensionless form of equation (1), we define dimensionless variables as
with the characteristic scales given by
where is a scale for the pressure buildup due to injection and is a scale for the lateral pressure propagation in the reservoir. Then, we can rewrite equation (1) in dimensionless form as follows
 The corresponding initial and boundary conditions are
where is the dimensionless well radius. The dimensionless parameter fields as a function of space are given by
 Accordingly, the dimensionless problem (9)–(11) has only three independent governing parameters defined as
where (m /s) is hydraulic diffusivity of the sandstone layer defined as
 Using (13), we can define another parameter, the ratio of hydraulic diffusivities , which can be expressed by and
 The effect of on the pressure dissipation in a layered system will be discussed in section 5.3.
2.4. Limiting Radial Solution
 If and either or approaches zero, the problem reduces to the solution of Theis , which is self-similar in the Boltzmann variable and given by
 At late time and large distance from the well, , Van Everdingen and Hurst  show that the pressure front is approximated by
 Previous studies of pressure diffusion have suggested that the self-similarity between and can explain the propagation of the pressure front in the radial semi-infinite domain [Van Poolen, 1964; Talwani and Acree, 1984; Shapiro et al., 1997; Daungkaew et al., 2000]. From the solution (16), it follows that the radial distance from the well to the pressure front, i.e., radius of review, given by
 The coefficient is defined by
where (Pa) is the cutoff amount for the pressure increase. In the limit of , the coefficient approaches 1.5 at which is not considered to be an independent parameter. In radial coordinates, the maximum dimensionless radius of review, , will be . Our numerical solutions to the radial problem show good agreement with the analytical solution (17). Later we show that dissipation into ambient mudrocks affects this first-order scaling law for the lateral propagation of the pressure pulse. Although the propagation follows a power law in all cases considered, the coefficient and, more importantly, the exponent decreases with increasing dissipation.
3. Numerical Results
 We perform finite-element flow simulations of the model defined by the governing equation (9), the boundary and initial conditions (10), for a large range of the dimensionless parameters (11) to study the effect of pressure dissipation on lateral pressure propagation. The analytic solution for the confined case (17) gives the maximum distance of pressure diffusion, and the computational domain was chosen much larger, so that the outer boundary conditions had no effect on the solution. The finite-element analysis is conducted with COMSOL Multiphysics  using bilinear quadrilateral elements for spatial discretization [Hughes, 2000] and a variable step method for time integration [Dreij et al., 2011]. We use numerical grids that are highly refined near the boundary of the reservoir as shown in Figure 5a to resolve the strong pressure gradient typical for this problem, as in the pressure field shown in Figure 5b.
3.1. Effect of Permeability Ratio
 Before considering the full problem governed by all three parameters , and , we illustrate the effect of pressure confinement or dissipation on the power law by varying while keeping and fixed at values typical for Cranfield-type reservoir, i.e., and . To compare multiple pressure plumes, we just show the pressure contour equal to the cutoff value corresponding to . Figure 5c shows how increasing mudrock permeability allows pressure dissipation and reduces the speed of lateral pressure propagation.
 Figure 6a shows that the propagation of the pressure front with dissipation into the mudrock remains linear on a log-log plot and is still given by a power law. To quantify these dissipative losses, we assume a general power law for the lateral pressure propagation of the form
where and are functions of , and . In the limit of impermeable surrounding layers, the exponent approaches 0.5 and the coefficient approaches . The exponent corresponds to the slopes in Figure 6a and decreases monotonically with increasing as more overpressure dissipates into the ambient mudrocks as shown in Figure 6b.
 The dependence of and on the dimensionless pressure cutoff is also shown in Figures 6b and 6c. Analogous to the radial analytical solution the power-law exponent in the layered problem is independent of the cutoff. The coefficient is reduced with increasing similar to the decay of given by equation (18). Figure 6c shows that the reduction of is strongest in the limit of , where approaches . As noted earlier, if is at its asymptotic value of 1.5 and similarly is constant.
3.2. Dissipation Parameter for a Layered System
 The scaling analysis presented in section 2.3 suggests that the power law (19) should be a function of the individual ratios of mudrock to reservoir permeability and specific storage ( and ) and the aspect ratio of the maximum radial diffusive distance to the reservoir thickness ( ). To analyze the effect of each parameter on the pressure evolution, we performed 480 simulations for a parameter study varying from to from to , and from to . These ranges for and correspond to the rock properties reported in the literature (Figure 3) and the range of corresponds to 30 year injection into a reservoir of thickness varying from 10 to 100 m. We define the location of the pressure front using a pressure cutoff so that , and the dimensionless location of the pressure front with time follows the power law in all simulations.
 The numerical results are summarized in Figure 7, which shows the variation of the coefficient and the exponent as a function of the governing parameters , and . Both and decrease with increasing , and , which implies that higher , and result in more retardation of the pressure front in a sandstone reservoir. In other words, more overpressure will be attenuated into mudrocks due to either the high permeability of the mudrock (large ) or the high specific storage of the mudrock (large ) or due to a large aspect ratio of the pressure plume, and hence a large surface area across which overpressure can leak into the mudrock (large ).
 Figure 7 shows that the contour surfaces of both and form a set of parallel planes in the three-dimensional parameter space of , and . The main result of this study is that the planar contours in Figure 7 show that the power law for pressure propagation depends on a single dissipation parameter defined by the normal to these planes. The normal is given by
and an arbitrary point in the parameter space. We define a dissipation parameter M that will collapse the three variables into a single one:
where ( ) is the arbitrary origin set to the minimum values of the governing parameters (−8, 0, 2) in this study. Figures 8a and 8b show that all data for and collapse to a single line if plotted as a function of M, and that M varies more than 2 orders of magnitude. Figure 8 shows that and are monotonically decreasing functions of M. Two plateaus, where the power law is a weak function of M, are separated by a sharp transition between M of 1.5 and 4. To obtain expressions for and that can be evaluated we fit numerical data in Figure 8 with generalized logistic functions
 Together with the expression for M in terms of the three governing parameters , and , equations (22) and (23) completely determine the power law for the lateral propagation of the dimensionless pressure.
4. Application to Storage
4.1. Reduction of the Radius of Review
 The radius of review can be calculated as the radius of the outer extent of the pressure front. Sample calculations of the radius of review are performed for injection with constant rate of 10 Mt of per year, which is equivalent to of per year, assuming density is 600 kg/m at average pressure P = 16 MPa and temperature T = 60°C [Bachu, 2003], into a sandstone reservoir surrounded by mudrocks for 30 years. For the cases considered below, a characteristic pressure is 200 MPa for In Salah-type, 8 MPa for Frio-type, and 5 MPa for Cranfield-type reservoirs in which varies up to 0.2 so that is in the range 1.2 to 1.5 as shown in Figure 8b. The In Salah-type reservoir has a large because of relatively low permeability of the reservoir . In this case, the maximum pressure near the well would have to be reduced by injection through multiple wells similar to Figure 1b or horizontal wells not to trigger hydraulic fracture. Injection through n suitably placed vertical wells would reduce the characteristic pressure to . Furthermore, in a two-phase flow system, i.e., injection into brine-saturated reservoir, the actual maximum overpressure will be lower than due to the higher mobility of phase as pointed out in Figure 4a.
 To illustrate the effect of pressure dissipation, we use the geometry and petrophysical properties of In Salah-type, Frio-type, and Cranfield-type reservoirs, summarized in Table 3. To facilitate comparison and to highlight the effect of pressure dissipation, we assume that the same total amount of (0.3 Gt equivalent to ) is injected at each site.
Table 3. Estimates of the Radius of Review and the Storage Capacity
The standard physical properties of each reservoir, i.e., and , are from Morris et al. , Hovorka et al. , and Choi et al.  given in Table 2.
 Using the data of each reservoir and the information of the injection operation as shown in Table 3, we can evaluate the dimensionless parameter M as defined in equation (21). The values of M vary from 1.85 for an In Salah-type reservoir to 3.19 for a Cranfield-type reservoir. Although the relative mudrock permeability is lowest at the Frio-type site, it has a higher M value and is therefore more susceptible to pressure dissipation than In Salah-type site, because the relative specific storage of the mudrock is higher and the aspect ratio of the confined pressure plume is larger. Both In Salah- and Cranfield-type reservoirs have similar geometry, i.e., is similar, but the M value of the Cranfield-type reservoir is much higher because of larger permeability and specific storage of the surrounding mudrocks. Larger permeability at the Cranfield-type reservoir represents an estimate of the effective value for the heterogeneous mudrocks.
 Given M for these sites, the sigmoid curve fit equations (22) and (23) give the coefficient and the exponent , and using the characteristic scales (8), the radius of review is given by
and is reported in Table 3. The Theis solution (17) gives the upper limit of the radius of review, , which regards ambient mudrocks as perfectly closed boundaries. Even this upper bound varies from 13 km at an In Salah-type site to 51 km at a Frio-type site, mostly due to much higher hydraulic diffusivity at the Frio-type site.
 If pressure dissipation into the ambient mudrock is included, all three injection sites see a significant reduction in the radius of review. The In Salah-type site with the lowest M value is least affected, but its radius of review after 30 years of injection is increased by 25%, if an impermeable and incompressible mudrock is assumed. As the M value increases, the use of overestimates the radius of review up to 58% for a Frio-type site. The Cranfield-type reservoir surrounded by highly permeable and compressible mudrocks results in a large reduction of the radius of review, and whether the enormous reduction in the pressure plume size at this last site is realistic is discussed in section 5.5. However, even in the other two cases, the reduction in the radius of review is large enough to significantly increase the storage capacity.
4.2. Estimating Storage Capacity
 In the section 4.1, we estimated the radius of review given a certain amount of injection. For the site characterization, the more relevant question may be how much can be injected, i.e., what is the storage capacity, given a maximum possible radius of review at a given site, .
 The retardation of the pressure front due to dissipation of overpressure into the ambient mudrocks needs to be considered in the estimate of the storage capacity. Solving the following nonlinear equation, we can estimate the maximum injection period at which the pressure front approaches the given from (19) as
 Given data for three sites, the amount of injected (Gt) can be estimated by
and is reported in Table 3. The results confirm that permeable and compressible mudrocks provide a larger storage capacity of the target sandstone reservoir. The In Salah-type reservoir with the lowest M value is the least affected by the ambient mudrocks, but the storage capacity is reduced by 46%, if an impermeable and incompressible mudrock is assumed. For a reservoir with a larger M value, the assumption of incompressible mudrocks underestimates the storage capacity up to 74% for a Frio-type reservoir. Similar to the estimates for the radius of review, the use of Cranfield-type properties leads to unrealistically large increases in the storage capacity that cannot be taken at face value and will be discussed in section 5.5.
4.3. Uncertainty Due to Mudrock Properties
 Figure 8 shows that the M values for all three sites are in the transition region (1.5 M 4). As a result, uncertainty in the physical properties leads to significant uncertainty in the radius of review and the storage capacity.
 The range of physical properties for mudrocks compiled in Table 2 and illustrated in Figure 3 is very large. This is partly due to the large variability of the rock types that are collected under this category in simulation studies. This reflects the difficulty of measuring representative properties, and in several cases, appropriate data are simply not available and the values are merely estimates. The combination of this variability and uncertainty in the mudrock properties with the sensitive dependence of the power law on M introduces large uncertainty to any estimate of the radius of review. If the storage capacity is limited by the lateral pressure propagation, the uncertainty in the mudrock properties will introduce large uncertainty into estimates of the storage capacity. The discussion of the uncertainty will focus on the In Salah-type and Frio-type sites, because as shown in sections 4.1 and 4.2, the physical properties assumed for a Cranfield-type site give unrealistic results.
 To illustrate this uncertainty, we generate 4000 values of permeability and specific storage of mudrock ( and ) from a lognormal distribution around the preferred values with a standard deviation of either 0.25 or 1. The resulting distributions of M values (M distribution) from the Monte Carlo method are shown in Figures 8d and 8e remain roughly lognormal, although some skew toward lower M becomes apparent as the variance increases. The effect that uncertainty in M has on the prediction of the radius of review depends on the location of the M distribution relative to the transition zone of the sigmoidal curves for and .
 The top row of Figure 9 shows the distributions of the predicted radius of review ( distribution), given a fixed injection volume and injection rate. As expected, the distributions become wider as the variance in the mudrock properties increases. The distributions for In Salah-type sites are both narrower and more skewed than for Frio-type sites. For both sites, the maximum of the distribution is centered on the mean for low variance, but becomes shifted at higher variance. These results can be understood in terms of the shape of the M curve. For the same variance in mudrock properties, the uncertainty in the prediction is larger for the Frio-type site, because the M distribution is located in the center of the transition zone, where the power law for the lateral pressure propagation is a strong function of M. The sigmoid shape of the M curve also explains the skew of the distribution for the In Salah site, where the low tail of the M-distribution samples the left plateau. As the variance increases, the number of realizations sampling the plateau increases until the maximum of the distribution is shifted to larger values. For Frio-type sites, the M distribution begins to sample both plateaus as the variance increases, leading to a bimodal distribution with a minimum near the mean value.
 The bottom row of Figure 9 shows the distributions of the predicted storage capacity ( distribution), given a fixed radius of review and a fixed injection rate. The shape of the distribution depends on the solutions of equation (25) and is more difficult to interpret. Again the uncertainty in the estimate increases with increasing variance in mudrock properties that makes uncertainty for the Frio-type site higher. For both sites, the distributions are centered on the mean and strongly skewed toward higher values.
 This simple analysis illustrates the complexity in estimating the uncertainty of the radius of review or the storage capacity. In particular, estimates based on the mean physical properties of the mudrock may be good guide, if the storage capacity has to be evaluated for a given radius of review, because the maximum of the distribution is close to the estimate based on the mean. In contrast, estimates based on the mean physical properties of the mudrock may not be a good guide, if the radius of review has to be evaluated for a desired storage capacity and if the uncertainty in mudrock properties is large, because the maximum of the distribution may be unrelated to the mean estimate.
4.4. Volume of the Displaced Brine Due to Injection
 The injection of fluid causes an increase in the storage formation pressure, which will induce the displacement of the preexisting fluid in the formation. For the two-phase flow system of injection into a brine-saturated reservoir surrounded by permeable and compressible mudrocks, injected will be trapped by capillarity, but the brine itself can leak into the ambient mudrock. In our single-phase flow model, we can calculate the amount of the fluid in each layer using the following equations
where (m ) is the volume of the fluid, and the subscript i is s for the sandstone reservoir and m for the ambient mudrock. The volume fraction of the total injected fluid that has been displaced into the mudrock is given by
 Figure 8c shows that data from the numerical simulations also collapse into a single line as a function of M. To obtain expressions for , we generate the generalized logistic function
 As shown in Figure 8c, increases with M because more fluids will be displaced into more permeable (larger ) and more compressible (larger ) mudrocks surrounding a thinner reservoir (larger ). For an In Salah-type site, approximately half of the displaced formation brine will be stored in the ambient rock and even more for a Frio-type site. This highlights that large fluid volumes can be absorbed by mudrocks due to their high specific storage during injection. Again, the Cranfield-type rock properties give unrealistic predictions. This suggests that the applicability of the simple hydrological storage model needs to be evaluated carefully.
5.1. Scaling Laws
 We have demonstrated that dissipation of pressure into ambient mudrocks can significantly reduce the exponent of the power law governing the lateral spreading of the pressure plume. In contrast, the nonlinearities of two-phase flow or phase change near the well do not affect the power law governing the radial pressure propagation [O'Sullivan, 1981] nor does the mechanical deformation of the reservoir due to full poroelastic coupling [Helm, 1994; Hsieh and Cooley, 1995]. In this sense, pressure dissipation into ambient formations has a more profound effect on the large-scale pressure evolution in the reservoir than these other processes that have received most of the attention in the literature on geological storage. The strong effect of dissipative terms on the power law for diffusive propagation is well known and has been discussed extensively in the literature on self-similar solutions. Barenblatt  has shown that the power-law exponent is a continuous function of the dissipative loss, in this case the parameter M, which correlates with the fluid loss from the aquifer.
5.2. Effect of Hysteretic Specific Storage
 The increase of pore pressure induced by injection results in a reduction of the effective stress, so that the target reservoir and the ambient mudrocks will be under unloading conditions. The response of sedimentary rocks to loading and unloading can be different, and this is usually described by the hysteretic specific storage given by Hueckel and Nova , Neuzil and Pollock , Neuzil , and Hoffmann et al. :
where and are the effective and virgin specific storage, and and σ are the effective and preconsolidation stresses. For fine-grained unconsolidated sediments, i.e., mudrock, is up to 2 orders of magnitude smaller than . Most studies of storage have neglected this hysteresis effect because geological storage targets deep, strongly compacted, and diagenetically altered formations, where hysteresis effects become smaller than in shallow groundwater systems [Settari, 2002]. However, some hysteresis is likely to occur and given the lack of detailed geomechanical characterizations of this adds additional uncertainty to the description of the mudrock properties. Qualitatively, the specific storage under elastic unloading is reduced by up to 2 orders of magnitude. This should reduce M values and shift them to the left plateau of the M-curve (1.58 for In Salah-type and 1.52 for Frio-type) at which both and are less sensitive to the uncertainty in M, but the dissipation into the ambient mudrocks will still account for 20% of the total injected fluids.
5.3. Hydraulic Diffusivity Ratio of Layered Systems
 Much of the literature on pressure diffusion [Terzaghi, 1943; Bredehoeft and Hanshaw, 1968; Talwani and Acree, 1984; Song et al., 2004; Jaeger et al., 2007] discusses one-dimensional transient-flow problems in a homogeneous domain where , and S can be combined into hydraulic diffusivity . The emphasis on hydraulic diffusivity in the literature may suggest that the ratio of hydraulic diffusivities would govern the lateral pressure propagation in this layered model. Typical values of are in the range of to , as shown in Figure 3.
 In heterogeneous media, such as the layered system considered here, equation (1) cannot be rearranged to introduce as a dimensionless parameter. Our results show that the power law for the lateral pressure propagation depends on instead of , and hence lines of constant are perpendicular to lines of constant M as shown in Figure 10a. This means that any and can be obtained from M for a given , i.e., the lateral pressure propagation is independent of the value of . We illustrate this surprising conclusion by comparing two sets of simulations:
 (1) Three simulations with a fixed value of , but a range of M values. Parameters are shown as circles on the solid line in Figure 10a, and the resulting pressure contours in Figure 10b.
 (2) Three simulations with a fixed value of M = 1.1, but a range of values. Parameters are shown as circles on the dashed line in Figure 10a, and the resulting pressure contours in Figure 10c.
 The pressure contours in Figures 10b and 10c show that simulations with constant have variable lateral pressure propagation but constant vertical pressure propagation, while simulations with constant M have constant lateral pressure propagation but variable vertical pressure propagation. The ratio of hydraulic diffusivities therefore controls the vertical propagation of pressure in the layered system while the dissipation parameter M determines the lateral pressure propagation.
 The observation that the lateral pressure propagation is independent of may be counterintuitive, if the amount of pressure that has diffused out of the reservoir is equated with the mass of brine that has leaked out of the reservoir. Equation (28) shows that the mass of brine displaced from the reservoir is proportional to the integral of the pressure times the specific storage of the ambient mudrock. Figure 10b shows that controls the vertical pressure profile but the mass of brine leaked from the reservoir also depends on the magnitude of the specific storage. Simultaneously changing and by the same factor does not affect the vertical pressure profile because is constant but it allows arbitrary changes in the mass of brine leaked from the reservoir. Section 4.4 also shows that the fraction of fluid mass leaked from the reservoir is only a function of M and therefore independent of . This clearly illustrates that it is the mass of brine leaked into the mudrock, not the distance of vertical pressure propagation, which determines the speed of the lateral pressure propagation in the reservoir.
5.4. Assumption of Single-Phase Flow
 The permeable and compressible ambient mudrock may cause a significant retardation of the pressure front, which may lead to interference between the retarded pressure front and the saturation front that has been neglected in this analysis. We can estimate the radial propagation of the plume, i.e., the saturation front, with time using a simple solution in Buckley-Leverett form [Woods and Comer, 1962]
where is the radius of -rich zone around the injector, is volume fraction factor for is gas saturation for , and is the fractional flow function that measures the gas fraction of the total flow given by
where is the mobility of the gaseous and the formation fluid ( and ), is the relative permeability that accounts for the reduced permeability of each phase due to the presence of another phase, and (Pa s) is the viscosity of each phase ( Pa s). The derivative of at the saturation of the front is determined by the tangent construction [Welge et al., 1962]. Bennion and Bachu  give relative permeability data for three sandstone and three carbonate formations that are candidate storage sites in onshore North American sedimentary basins. From this data set, the range of the derivative of at the tangent point is .
 The corresponding range in the location of the saturation front after 30 year injection is shown as the gray band in Figures 9a and 9b. For both In Salah- and Frio-type reservoirs, the retarded pressure front corresponding to the mean mudrock properties propagates much further than the saturation front , and the single-phase approach gives a reasonable estimate of the radius of review even with the attenuation of overpressure into the ambient mudrock. If uncertainty in the mudrock properties is large , the low end of the distribution is comparable to the location of the saturation front, and the pressure front interacts with the saturation front. In these cases, the location of the pressure front in the two-phase model is likely to be larger then the estimate based on single-phase flow.
5.5. Limits of the Transient Flow Equation
 The transient flow equation (1) is a special case of the more general theory of poroelasticity in the limit of uniaxial strain and constant vertical stress. In hydrologic applications, the difference between them is usually small. Linear poroelasticity itself, however, is a linearized theory that is only applicable for small deformations. In section 4, the Cranfield-type physical properties led to very strong retardation of the pressure pulse and corresponding very large storage capacity. High specific storage of the ambient mudrock allows large volumes of fluid stored in the mudrock, in this case, more than 90% of the injected fluid volume. In reality, however, such large changes in fluid storage are likely to lead to significant deformation that is not accurately described by constant specific storage in a linearized theory.
 Accurate predictions of the radius of review and the storage capacity in aquifers surrounded by very compressible mudrocks may therefore require nonlinear geomechanical models that account for changes in specific storage with increasing deformation. Currently, most estimates of the pressure buildup are based on linear theories, and the results need to be evaluated carefully if M is high.
6. Summary and Conclusion
 This study shows that ambient mudrocks can have a strong impact on the pressure evolution within a geological storage reservoir. The lateral propagation of the overpressure due to injection is significantly reduced by dissipation into ambient mudrocks.
 In a layered system, the pressure evolution is governed by three dimensionless parameters defined by ratios of permeability and specific storage between the reservoir and the ambient mudrock ( and ) as well as the aspect ratio of the confined pressure plume ( ). Numerical simulations show that the spreading of the pressure front with time in the sandstone reservoir follows a power law even with attenuation of overpressure into the ambient mudrock. The coefficient and the exponent in the power law are governed by a single dissipation parameter . Both the coefficient and the exponent of the power law are sigmoid decreasing function of M. The ratio of hydraulic diffusivities does not describe the lateral pressure propagation, but it is suitable to describe the vertical pressure propagation in the layered system.
 A compilation of physical mudrock properties used in recent studies of geological storage shows large variability in the three governing parameters and hence in the dissipation parameter M. This suggests that the importance of pressure dissipation into the mudrock can vary widely between different storage sites. The power law described here enables the quantification of the uncertainty in predictions of the radius of review and the storage capacity using the Monte Carlo method. The M values of typical geological storage sites are in the region where the power law is changing rapidly. In combination with large uncertainty in mudrock properties, this leads to wide and strongly skewed probability distributions for the predicted radius of review and storage capacity.
 Our results show that estimates based on the mean physical properties of the mudrock may be a good guide if the storage capacity has to be evaluated for a given radius of review. In contrast, estimates based on the mean physical properties of the mudrock may not be a good guide if the radius of review has to be evaluated for a desired storage capacity. In this case, the mean values are likely to underestimate the radius of review when the uncertainty in mudrock properties is large.
 The characterization of the physical properties of the mudrock is an important component of the evaluation of a geological storage site. The effect of pressure dissipation into the ambient mudrocks should be considered in simulation studies of geological storage that aim to determine the pressure evolution.
 The authors are grateful to the sponsors of the Gulf Coast Carbon Center, Bureau of Economic Geology, The University of Texas at Austin, which partly supported this work and to its Director Susan D. Hovorka. Additional support comes from EPA STAR grant R834384 and the SECARB project, managed by the Southern States Energy Board and funded by the U.S. Department of Energy, NETL, as part of the Regional Carbon Sequestration Partnerships program under contract DE-FC26-05NT42590. The authors gratefully acknowledge anonymous referees of this paper for their constructive comments as well.